Download Developing Mathematical Thinking

MEYL624
Module 2
The Open University,
Centre for Mathematics
Education
Mathematics/Education
MEYL624 Interfaculty Course, Level 3
Developing Mathematical
Thinking
and the
Esmée Fairbairn Foundation
Module 2: Targeting thinking
MEYL624
Module 2
Mathematics/Education
MEYL624 Interfaculty Course, Level 3
Developing Mathematical Thinking
Module 2: Targeting thinking
Contents
2.1
2.2
2.3
2.4
2.5
Introducing individuals
Processing problems
Reasoning realistically
Constructing connections
Communicating concepts
1
2
4
6
9
Holton, Derek (1993) What mathematicians do
and why it is important in the classroom,
Set number 1, 1993, Item 10, pp 1-6
12
Poster 1: Cutting a square into 11 squares
21
Reading 2
Appendix
The Open University
Centre for Mathematics Education & The Esmée Fairbairn Foundation
Developing Mathem atical Thinking at Key Stage 3 Course Team
Barbara Allen, Course Chair, Author, Open University
Gaynor Arrowsmith, Critical Reader, Open University
Andy Begg, Author, Open University
Alan Graham, Author, Open University
Linda Haggarty, Critical Reader, Open University
Maggie Jenkins, Course Manager, Open University
Peter Johnston-Wilder, Author, Open University
Sue Johnston-Wilder, Author, Open University
John Mason, Critical Reader, Open University
Cheryl Periton, Critical Reader, Open University
Pat Perks, External Assessor, University of Birmingham
Acknowledgem ents
We are grateful to the Esmée Fairbairn Foundation who provided a generous grant for the development
of the courses Developing Mathematical Thinking at Key Stage 3 (MEXR624, a 10-credit one-week
residential course), and Teaching Mathematical Thinking at Key Stage 3 (ME624, a 20-credit 26-week
distance course).
We acknowledge with thanks, the work of Bill Norman of the Materials Production Unit of the Open
University who prepared the artwork for the modules.
Acknowledgement is made to the Cockcroft report [Cockcroft, W. H. (1982) Mathematics counts,
London: Her Majesty’s Stationery Office], and to the advancement of technology, because these two
changes have inspired many mathematics educators in the quest for better learning tasks over the last
twenty years. Many of the learning tasks they devised were modified from other sources and it is very
difficult to source the original innovators. In designing this course the authors have openly drawn on
this rich tradition.
The Open University, Walton Hall, Milton Keynes, MK7 6AA
First published 2002, Revised 2003, Revised 2004
Copyright © 2002 The Open University
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or
transmitted, in any form or by any means, without written permission from the publisher or a licence
from the Copyright Licensing Agency Limited. Details of such licences (for reprographic
reproduction) may be obtained from the Copyright Licensing Agency Ltd, 90 Tottenham Court Road,
London W1P 9HE.
Printed in the United Kingdom by The Open University.
For further information on Open University courses, write to:
Central Enquiry Service, PO Box 200, The Open University, Walton Hall, Milton Keynes MK7 6YZ.
ISBN 0 7492 5792 X
1.2
2.1
Introducing Individuals
The aim of this module, Targeting Thinking is to focus attention on the thinking skills that make up
‘mathematical thinking’. This includes looking at different ways of tackling the same problem,
addressing connections both within mathematics and outside it, and considering how to improve the
ability to communicate mathematical ideas by working on powers of mental imagery.
Task 2a
i
Introductions
Fold a piece of A4 paper into three to form a triangular prism.
Put your name on it, and underline what you would like to be called.
Now put it in front of you so we can all see it.
Which way did you fold the paper?
What difference did it make to the area for your name?
What difference did it make to the volume of the prism?
ii
With another piece of paper, (and sticky tape), what other shapes could you make
for your ‘nameplate’?
iii
Students and tutors to stand up and introduce themselves. Each introduction should
include at least one reference to mathematics (including examples, attitudes and
feelings).
One “big idea” that links many of the themes within ‘mathematical thinking’ is problem solving. But,
before you explore this notion you need to have some shared notions about what mathematics is—that
is, about what mathematicians know and do.
Task 2b
Knowing and doing
The ‘cards’ below represent tasks suitable for primary pupils. Sort them into two
categories—those tasks that test facts and skills only and those that require
mathematical thinking (rather than merely following procedures).
What are the prime factors of 209?
Do the following addition: 5 + 6 + 3 + 17 =
Show why (a + b)(a – b) = a2 – b2
Name three fractions between 8/10 and 9/10.
What is the length of this card?
What numbers are the same as their squares?
Solve the equation 4x + 7 = 399
When do you use a median to summarize data?
What is the 16th term of 1, 3, 5, 7, 9, …
Find the area of a triangle with sides 10, 24, and 26cm.
What is 140% of 45?
(1, 3), (3, 2) and (6, 0) are points on a line, T or F?
What are three other ways of writing ½?
What is the sum of the angles inside a pentagon?
Can you tessellate with pentagonal tiles?
Prove that the sum of first n odd numbers is n2.
Can the area of a rectangle equal its perimeter?
How would you explain how to bisect an angle?
What is the total surface area of this room? A triangle has sides 3, 4, and x cm. What might x be?
1
£1 = €1.6. How much is €8 worth?
Can the mode, median and mean of a sample be the same?
Can you cut a sphere to get an ellipse?
3 × 11/2 = 3 + 11/2. Find other pairs of numbers like this.
Is 1 billion the same as 106 or 109?
Tossing dice, is two 6s harder to get than two 3s?
Divide 150 by 25
Find x if 2, 3, 5, 7, and x have a mean of 5
What is 3 + 4 × 2 + 1?
Find angles on a pie chart to represent frequencies 1, 2, 9.
Draw a net for a 2-person hiker’s tent
What might the equation y = 14x + 70 represent?
2.2 Processing Problems
An important feature of ‘mathematical thinking’ is problem solving. Clearly, there are many different
types of problems as well as a range of vocabulary associated with problem solving.
Task 2c
Types of problems
The first part of this task is to consider problems i – x, so work on them briefly.
Then in xi – xiii you are looking for common elements in these problems.
i
A car goes from one town to another at 90 km/h and returns at 60 km/h. What was
the average speed of the car?
ii
How many first class stamps can you buy for £5?
iii
A bus carries 47 passengers. A school wishes to transport 210 students and 7
teachers on a school trip. How many buses are needed?
iv
The sum of two whole numbers is 17. What is the greatest product possible?
v
How many heads will you get if you toss a coin six times?
vi
Show that the sum of the angles of a triangle equal 180º.
vii
A gardener makes compost from loam, peat and sand in the ratio 7:3:2. If she had
1.5 litres of peat, how much loam and sand would she require?
viii
Design a bookcase to hold 200 paperbacks and 20 CDs.
ix
Find when 6 + 8 = 2.
x
Find x when 2(x + 3) = 2x + 6?
Next, having worked on the ten problems above, consider the following:
xi
What common elements do they share?
- Which of the questions required you to change from a real world context to a
mathematical one?
- Which used a simple mathematical procedure?
- Which ones required you to check and or change the solution to fit the real world
situation?
xii
Which of these problems are ‘theoretical’ mathematics problems, which are
practical or applied problems that model real situations, and which are design
problems where more information is likely to be needed.
2
xiii
Can you classify these problems into: no answers, only one answer, a small number
of answers, many answers, and always true.
Task 2d
Alternative solutions
A farmyard. contains both chicken and sheep. The farmer knows there are 26 heads and
74 legs. How many chickens and how many sheep are in the yard?
Solve this problem. When you have an answer, check it using another method, and then
see if you can find a third method to solve this problem, and another, …
Then share the ways for solving this (and similar problems) with the group.
What are the similarities and differences between these ways?
Task 2e
11 squares
Can you cut a square into exactly 11 squares with no scraps and no overlaps?
Discuss your results with others at your table.
Think about the following
- Is this possible?
- If it is possible, is there more than one way?
- Is it possible with only two different-sized squares, ( and three, four, ...)?
- What numbers other than 11 could be used?
Task 2f
???
3
2.3
Reasoning Realistically
Reasoning is an important part of problem solving, in particular with theoretical problems, but also in
terms of explaining to others, and in justifying one approach in preference to another. Reasoning does
not always have a formal logical pattern, and it often helps to be informal.
Questions that come under the heading ‘reasoning’ include the following.
What is it that is (or seems to be) true?
How can I be sure? (What kind of reasoning might I use?)
Why is it true?
Can I convince someone else that it is true?
These are standard questions in mathematics that you are likely to ask as you move from making an
initial conjecture to verifying or proving when looking at patterns or formal results.
Task 2g
Being certain?
i
Is the following diagram a proof of Pythagoras’s theorem?
ii
Is the following diagram a proof that 64 = 65?
Mathematical statements may be ‘always true’, ‘always false’, or ‘sometimes true’. When a
mathematical statement is claimed to be ‘true’, this means ‘always true’. To disprove a ‘true’ statement
you only need to find one counter-example. Correspondingly, to disprove a ‘false’ statement you need
to find one example of it being true.
Task 2h
Always, sometimes, never?
i
What is a prime number?
ii
‘If n equals 1, 2, 3, … , then 2n is never a prime number.’
Is this a true statement?
iii
‘Consider 6n + 1. When it is not a square, it is always a prime number.’
Try n equals 1, 2, 3, … to find whether this statement is true or false.
4
iv
‘If n equals 1, 2, 3, … , then n2 – n + 11 is always a prime number.’
Is this true or false? (Justify your answer.)
v
‘1 is a prime number.’ True or false? (Why?)
vi
‘All squares are rectangles.’ True or false?
vii
‘All rectangles are squares.’ True or false?
viii
‘If a + b = 1, then a + b2 = a2 + b.’ Is this true? (Give reasons.)
5
2.4
Constructing Connections
‘Constructing connections’ includes making connections within mathematics, across subject
boundaries, with everyday life, and with tools and technology.
Constructing connections is important in learning. It is the process by which learners make sense
of their worlds. ‘Sense-making’ occurs as ideas are linked into increasingly complex but coherent
mental structures (or ‘knowledge schemas’).
While numerous tools to aid calculation have been used in mathematics in the past, electronic
calculators and computers are relatively new. However, using these is much more than automating
arithmetic. Technology is useful in all areas of mathematics, and it provides tools for teaching and
exploration.
Task 2i
Scientific calculat ors
i
Enter a three digit number into your calculator.
Now see if you can reduce the number to zero with only five operations using
any of the four functions but using only single-digit numbers for each step.
For example, starting with 435 the five steps might be, – 3,  6,  6, – 9, – 3.
ii
Imagine the 8 and 9 keys on your calculator were broken. How could you add
398 and 829 on this calculator without using these two keys? How many
different methods could you use?
iii
Explore ! on a calculator, what does it do?
What if the number you use ! with is negative, zero, or a fraction?
When is the result greater than the starting number, when is it less than, and
when is it equal to the starting number?
iv
What happens if you use  on a number other than 1, 4, 9, 16, 25, … ?
What happens if the number that you use is negative, a fraction, or zero?
When is the result greater than the starting number, when is it less than, and
when is it equal to?
What key is the inverse of  ? (That is, what undoes  ?)
Is the square root of a number always positive?
v
What does 1/x do on the calculator?
(Note, some calculators label this button as x–1 )
What if the number you use 1/x with is negative, zero, or a fraction?
When is the result greater than the starting number, when is it less than, and
when is it equal to the starting number?
What key ‘undoes’ 1/x ? (That is, what is the inverse of 1/x?)
vi
What does xy do on a calculator? (Start with x3, x4, …)
What happens if either x or y is zero, negative, or a fraction?
When is the result greater than the starting number, when is it less than, and
when is it equal to the starting number?
Does the value obtained depend on your calculator?
vii
What advantages or disadvantages would you see in using calculators with
your class? How does the use of calculators promote or discourage
mathematical thinking?
6
Task 2j
????
Task 2k
????
7
2.5
Communicating Concepts
Communication is usually conducted through the medium of words (spoken and written) and in the
context of mathematics words are extended to include symbols. Mathematical communication also
involves pictures, diagrams, and three-dimensional models. A powerful aid to creating and interpreting
visual displays is to draw on pictures from within your own imagination—i.e. by exercising your
powers of ‘mental imagery’.
Task 2l
i
A thought experiment
Try this task first as a ‘thought’ experiment; that is by ‘mental imagery’.
Then do it with a sketch, or using real counters or cubes.
How many counters need to be moved to change from
this shape
to
this shape
o
o o
o o o
o o o o
o
o
o
o
o
o
o
o
o
o
ii
What is the least possible number of moves?
Try with smaller and larger triangles.
iii
Does a pattern emerge?
Task 2m
o
o o
o o o
o o o o
More thought experiments
Here are some more ‘thought’ experiments. Try each of them first in your head,
then check with a diagram, and if unsure, check with solid cubes. Discuss your
thinking with others in your group.
i
Imagine a pile of wooden cubes and one pot of paint.
Each face can either be painted or left unpainted.
How many differently painted cubes are possible?
ii
Imagine a 3  3  3 cube made of 27 small unpainted cubes.
The outside of the 3  3  3 cube is then painted
After the paint is dry it is dissembled.
How many of the small cubes will have 0, 1, 2, or 3 faces painted.
How does this pattern change for 4  4  4, and 5  5  5 cubes?
Can you get a general rule for a n  n  n cube?
iii
I have eight cubes: two red, two blue, two yellow, and two green.
I want to make a larger cube with them so each colour appears on each face.
In how many ways can I make this large cube?
8
Reading 2 Holton, D (1993) What mathematicians do and why it is important in the classroom, Set
number 1, 1993, Item 10, pp 1-6.
Note: This article was published in 1993, too late to include mention of Andrew Wiles’ successful
proof of Fermat’s Last Theorem, which he completed in June of that year.
What mathematicians do and why it is important in
the classroom
Derek Holton
University of Otago
The only necessary prerequisite for reading this article is that, at some time or other, you have failed to
balance your cheque book.
1. What do mathematicians actually do?
I have wondered for years why more students don’t enjoy mathematics. I spend, and enjoy, a great
many hours each week doing something which is, apparently, anathema to almost everyone else. Why
is this so?
Leaving aside obvious answers about my twisted personality, I think the problem is that
schools and universities usually teach only a small part of what mathematicians do. I suspect that the
most exciting part of mathematics is too often kept out of sight of children. So let’s see what research
mathematicians do.
1. New Problem s
First, mathematicians look about for new, worthwhile, unsolved, problems; mathematics is not all sewn
up, done, and finished. There are in fact, more open problems today than there have ever been before,
but it is very difficult to explain most of them to a layman. These problems require too much
preliminary mathematics; to even understand the question, you often need to have studied for many
years.
But there are still some problems that can be easily explained. One of these is the so-called
Fermat’s Last Theorem. He wrote in the margin of a library book that he had a proof of the result ‘but
the margin is too small to contain it’, I hope he was banned from the library! He should certainly have
been banned from the mathematicians’ guild: his tantalising scribble was made about 1650 and there is
no known proof of the result even today.
So what is the problem? Well, most people know Pythagoras’ Theorem. The square on the
hypotenuse (the long side) of a right angled triangle equals the sum of the squares on the other two
sides.
Algebraically we can write this as x2 + y2 = z2, where z is the length of the hypotenuse and x
and y are the lengths of the other two sides. There are whole numbers (integers) which we can use
instead of x, y, and z and end up with a true statement. These are called solutions to x2 + y2 = z2. The
famous 3, 4, 5 triangle used by builders from ancient Egypt to the present day to get a right angle, is
one solution since 9 + 16 = 25 (32 + 42 = 52).
But a sort of 3-dimensional version of Pythagoras doesn’t seem to work: (3 3 + 43  53); 27 +
64 does not equal 125. Does it ever work? What about x4+ y4 = z4? Does that have integer (whole
number) solutions for x, y, z?
As far as we know xn + yn = zn has no solutions in which x ,y and z are whole numbers once n
is greater than 2. But no-one has been able to prove this, despite Fermat’s claim in the margin. So, there
9
are still unsolved questions in mathematics. What else do mathematicians do, after they have found a
problem?
2. Experim entation
When faced with a problem like that of Fermat, that can’t be solved immediately, our mathematician
plays around with it. In the case of Fermat’s problem, the natural thing to do would be to try n = 3 and
see if there are some integers x, y, z for which x3 + y3 = z3.
Having shown that there are no integer solutions in the case of n = 3, the next step is obvious. Test
n = 4.
3. Conjecture
After some time working this way, the mathematician would be forced to conjecture.
C.l. For whole numbers n bigger than 2 you cannot get a whole number answer to xn + yn = zn.
The conjecture has firmed up the problem (if it hadn’t been firmed up from the start). Most
frequently this is where the mathematician’s experience and intuition come into play. Guessing the
right conjecture is almost always the key step in finding new results.
4. Proof
The conjecture needs to be established as true, or condemned as false. We need a proof or a counterexample: just one case which doesn’t fit the conjecture dooms it to failure.
And now the mathematician is on a see-saw. Try to find a proof. If you can’t get all the way
with a proof, find a counter-example. If you can’t find a counter-example, try to find a proof following
a different route. How this cerebral process works is anyone’s guess. But generations of
mathematicians have made progress this way. Even when a proof is found, the mathematician’s work
still isn’t done.
5. Generalise
Suppose we have proved the conjecture C.l. How might we generalise or extend it? A generalisation
might be
G.l. For whole numbers n bigger than 2 there are no fractions x , y, z such that xn+ yn = zn.
Since whole numbers are fractions (5 = 5/1, etc.) Gl is a genuine generalisation of Cl. That is, Cl is a
special case of Gl. Proving the generalisation G.l. would immediately settle the Fermat Conjecture. So
far, it hasn’t been done.
An extension is a result which somehow grows out of the first result but is not a true
generalization. An extension of Fermat’s Conjecture is
E.l. For whole numbers n greater than 2 you cannot get a whole number answer for
xn+ yn + zn = un .
Now we are looking at four integers. So it is an extension rather than a generalisation. (By the way,
even if you could get an answer, that would not completely prove the Fermat Conjecture, unless z can
be zero.)
Mathematicians progress and develop a whole field of knowledge by this process of
generalisation and extension. It is worth looking into any maths textbook to see how this process has
worked there.
Although in the final published version, results are presented in large steps, they are usually
developed in very small steps. Settling a host of little conjectures is usually the only way to achieve one
reasonable-sized conjecture. It’s often considered that knowing the right questions to ask, that is,
making the right small conjectures, is really the heart of mathematics. A good mathematician is one
who knows the right questions to ask.
Having developed a whole theory though, the mathematician is back at step one, looking for a
problem. But if our mathematician is proud of the whole theory she or he will certainly want to spread
it around.
6. Publication
Mathematicians don’t accept every problem as worthwhile. Generally speaking, a result which doesn’t
have any generalisations or extensions which are of interest, is not thought worth spending research
time on. Problems set for the International Mathematical Olympiad competition for secondary students
are often in this category. They are not trivial questions but they don’t lead anywhere; they are good for
learning the mathematician’s trade but they don’t push mathematical frontiers forward.
10
What is considered a good problem may also be a matter of fashion. For instance, Gauss, one
of the all-time-greats, worked on non-Euclidean geometry but did not publish his results because he
thought other mathematicians would ridicule him.
Having solved what we think is a nice little problem, one which makes some contribution to
an area of mathematics, we now have to write the problem up and hope that it will be published in
some mathematical journal. For a Pure Mathematician this usually involves the statement of a theorem,
its proof, finished off perhaps by a conjecture or a remark as to where the new result fits in to existing
mathematical knowledge.
In this writing-up, just like a textbook, there is frequently no hint of all the trouble that went
into the creation of the final work. Like a birth certificate it guarantees the existence of an individual
but shows none of the effort (and fun) that its parents put into its creation, nor the attentions of the
medical staff that helped bring it into the world.
Why do we publish such bare-bones information in Journals and textbooks when the really
exciting part of mathematics is the creative part? Why do we show our students only the birth
certificates? Is it embarrassment?
Once a paper has been submitted, the Journal editor passes the paper on to one or more
referees. Is the paper correct? Have we indeed proved our result? And is the result sufficiently
interesting to publish? If our paper passes the referees’ tests, in due course it will appear in the journal
and we will be provided with reprints that we can send to our friends to show them how clever we’ve
been. We can put the paper in our curriculum vitae and, if we publish enough of sufficient quality, then
we’ll be promoted to Senior Lecturer or even Professor. It may even happen that our paper will spur
someone else on to a generalisation or an extension and eventually solve a really major question.
Even when you think you’ve got it right though, the referee may find an error. Worse still, and
this happened to me recently, even after the paper has been published, someone reading it may find a
mistake. (Now that is embarrassing!)
Mathematics is not necessarily a game for hermits or recluses. Although many mathematicians
still do work on their own, research papers are more and more often the product of two or more people.
I personally find that joint research is more enjoyable and leads to fewer suicidal feelings when things
are going badly. Usually too, results come more quickly because different people bring different
experiences to bear on the problem in hand. And surprisingly ‘talking’ mathematics makes it clearer
than mulling it around in your head. So mathematics does not have to be a game for one player.
I know very few mathematical research workers who don’t enjoy what they’re doing for its
own sake – for the sake of treading where no-one previously trod. As a result, most of us spend far
more than 40 hours a week on the job, for fun.
The Seven Steps of Mathematical Research
Step 1. Find a problem
Step 2: Experiment
Step 3. Conjecture
Step 4. Proof or counterexample
Step 5. Generalise or extend
Step 6. Publish
Step 7. Go to Step 1
2. What is mathematics?
I’m never totally sure that I know the answer to that question. However I will try to say not what
mathematics is, in total, but rather what I think the most important ingredients are.
1. Objects
To me the most fundamental part of mathematics are its objects. These are the numbers, sets, algebraic
quantities, graphs and so on, that are the basic items of study. Anything in the real or an imagined
world is potentially an object for mathematics.
For instance, take points and lines. We think of them as real and use them to construct plans of
houses, roads, aircraft, machines … which can be subjected to the rules of geometry as required.
However, points and lines are imaginary objects in space; and that space is the one imagined by Euclid,
which approximates to the real world on many occasions, but not all.
11
2. Algorithm s
The second basic ingredient of mathematics are the rules of the particular game we are playing. These
allow us to manipulate objects, kick them around, score goals if we’re lucky. Called algorithms, they
are well defined processes which do a particular job. A good algorithm will perform a particular
function in a finite number of steps. Algorithms are almost always able to be converted into computer
programs. All computer programs are algorithms.
The common algorithms that everyone has used in school are multiplication and division. The
multiplication algorithm allows us to multiply two numbers. For instance, 24  17 is calculated as
follows:
24
 17
168
+ 240
408
The division algorithm, in one form, works like this:
24
17) 408
– 34
68
– 68
00
Although not often called an algorithm, we can think of the process of solving a linear equation as an
algorithm. We illustrate the finite number of steps used in the example below.
Solve
3x – 6 = 0
3x
= 6
x
=
6
3
x
= 2.
The algorithm implicit in these steps will solve any linear equation.
I regard mathematical formulae as algorithms because they give a well-defined method for
solving certain problems. Hence A = r2 an algorithm in the sense that it says, if you give me the radius
of a circle, I will give you (in a finite number of steps) its area.
Similarly x 
 b  b 2  4ac
is an algorithm for solving the quadratic equation ax2 + bx + c = 0.
2a
3. Theorem s and Logi c
At a slightly higher level in the mathematical hierarchy we have theorems and logic. Logic acts on
theorems in much the same way as algorithms act on objects. But, so far, it is not as easy to programme
logic to manipulate theorems as it is to programme algorithms to manipulate objects.
Given an object I can perform an algorithm on it. Given a known result, it can be proved by
following a sequence of logical steps. There is nothing original here. We just follow the train lines from
one station to the next. There is nothing unexpected; we know the algorithm will produce the right
result; we know that logic will correctly prove the next theorem.
4. Creativity
How does creativity fit in? Creativity produces new discoveries. It extends our list of theorems,
algorithms and logical tools. It gives us more control over the world we live in.
Creativity produces problems which lead to conjectures. It enables us to find counterexamples to conjectures, or proofs for new theorems. It is creativity that leads to generalisations and
extensions and provides connections between previously unrelated sections of the subject. We know
creativity when we see it, and we know what it does. How it does it is still locked inside the brain.
In the future it may be possible to program a machine to be creative. But this creativity will
have to be more than applying logic to one theorem to produce new theorems. This is because most
theorems produced in this way will not be useful, nor will they be interesting. Part of the creativity of a
mathematician is to be able to see what is potentially useful and interesting and link that to the
mathematics that is currently known.
So mathematics consists of certain planks, and tools – the objects, algorithms, theorems and
logic. But mathematics also involves an imaginative side, the creative part that enables new tools to be
invented and the wood to grow and to be of more use; that produces results that have not been thought
12
of before; that produces processes to be developed, that can’t be found in any textbook or mathematical
paper; that is the most interesting and satisfying part of the whole subject.
3. What to teach?
1. Being Creative
We are extremely good at exploring the objects of mathematics and at teaching any number of basic
algorithms. At university we show our students endless theorems and often expect them to learn the
proofs.
I first got fun from mathematics because I could continually get the right answer. It was not
till later that I realized that there was a completely different side to it all. It was possible to produce
something entirely new. My entirely new results were rather like finding a dead tree in the Opaharua
Swamp. I had to (like Newton) stand on the shoulders of giants to do it. But I was seeing things that noone had seen before. It didn’t matter that it was no more useful than a dead tree. It was my dead tree.
And I kept hoping that, if I kept looking, there might be a swan behind the dead tree.
In recent years we’ve started to move towards more creativity in the mathematics classroom.
This has been achieved to some extent by the introduction of problem solving. Students get a chance to
take part in a search and sometimes get the feeling of discovery and creativity. Problem solving is the
closest thing I know of at the school level, to real mathematical research. With good problems, students
can go through all of the creative procedures: they can conjecture, find counter-examples, find proofs,
generalise, extend, and even publish. In 1952, Ann Roe studied 64 eminent scientists. She found that
the single most important factor in the final decision of these people to take up science, was the sheer
joy of discovery. With problem solving we are giving children a taste of that joy.
There is a small bit of cheating involved here though: (i) the students generally do not dream
up the problems for themselves, and (ii) the answer is already known to someone before the problem is
posed. But it is a big step in the right direction.
2. Facts, and Using Th em
The human race is storing up more and more facts each day. Probably they are being accumulated at an
exponential rate. How can anyone even expect to learn them all? Luckily knowing facts is not as
important as being able to use facts, and to use them in new and novel ways. It is necessary for our
students to learn to think deductively and intuitively. We need to give them practice in creativity.
This is not to say that they should learn no facts. Of course, students need to know
mathematical facts. They need to know about mathematical objects, algorithms, theorems, and logic.
But they should also know how to generate ideas.
The reasons for this are because (i) creativity is an integral part of mathematics (and all other
disciplines); (ii) we are trying to prepare our students for a world which will not be like today’s world –
possibly the best training we can give them is to be flexible, to think and to be creative.
There are clues from educationalists about how this may be done. Joseph Renzulli in his The
Enrichment Triad Model has three types of enrichment which can lead students to genuinely creative
experiences. I’ll take up his themes in the next section, though they have already appeared implicitly.
The only problem is likely to be a need for teachers to change their attitude – it is almost certain that
some students will get away, will be off and running, so fast that teachers won’t be able to keep pace
with them. I often have this problem. Bright school students I’ve worked with and my graduate
students often leave me behind. You just have to relax and enjoy it. Teachers should encourage their
students to try out new ideas even if it means disrupting school timetables – perhaps for them to go to a
library or visit an expert. And of course, not just for the study of Mathematics.
4. Renzulli was there first
Renzulli is an educator who developed his ideas through an interest in talented students in all areas, not
just mathematics. At the heart of his programme are three kinds of enrichment activity – Types I, II and
III. I have adapted his general ideas to the teaching of mathematics, and to the teaching of all pupils,
whether obviously talented, or not.
Type I
Type I enrichment activities are general exploratory activities: the teacher provides experiences that put
students in contact with topics or areas of study in which they may develop a sincere interest. Although
the emphasis is on exploration, the students need to realise that it is purposeful exploration. Eventually
students will be expected to conduct further study in one of their areas of interest.
13
Renzulli suggests 3 ways for students to explore: through interest centres, field trips and
resource people. You may well be able to think of others. A mathematics interest centre could contain
a range of books containing mathematics not met in the regular classroom, bibliographies of famous
mathematicians, puzzle books, popular books on mathematically related topics, etc., appropriate to the
level of the student. They could also contain a range of equipment. Apart from the obvious items,
maybe you could find a theodolite, an astrolabe or some other exotic equipment. (At my last university
I was custodian of three magnificent machines from the last century that variously found the arclength
of a curve and constructed conics.) It’s worth keeping your ear to the ground. It’s amazing what
equipment universities and companies discard. It’s worth visiting antique shops too from time to time.
Probably interest centres could also include these days, a computer and software. For example, there is
some very nice fractal and chaos software available now.
It is fairly obvious what is meant by field trips. There are an increasing number of hands-on
science museums springing up. These all have some mathematical items. My university encourages
school classes to visit the university and I’m sure others do too. In Britain, I know that some airports
are willing to take groups of students. That may be the case here as well. The Meteorological Office, a
polling firm, an engineering design office, a standards testing lab, an actuary’s department, a city or
county council engineering branch, an economic forecasting firm etc., are all possibilities. The aim is to
find local places where mathematics is actually being used in some form or other and see if they’ll let
you and your students in for a good look round.
Resource people could be friendly bearded professors, or people in government or industry.
It’s good for the students to see live mathematicians. Try to cultivate one or two that you know have a
good story to tell or some good problems for the students to try.
In summary then, the Type I activities aim to broaden students’ knowledge and experience by
placing them in situations which are different to the regular classroom routine. It is hoped that this
process may give students some particular interests that they will follow up to the Type III level.
Type II
Type II activities are almost exclusively training exercises. But these are definitely not contentoriented. Hence Type II activities are meant to lead students to a mastery of the processes which enable
them to deal more effectively with content. Learning how to use a formula or algorithm is not a Type II
activity; but learning to select the right algorithm is. These activities include critical thinking, problem
solving and brainstorming.
They should be planned so that they provide opportunities for developing thinking and feeling
abilities to their highest potential. They should also give introductions to more advanced kinds of study
and enquiry. It is hoped that they will provide skills and abilities to solve problems in a variety of areas
and in new situations.
There are any number of mathematical problems and projects that students can undertake, on
their own and in groups, that will facilitate Type II processes. These activities should highlight the
processes at work, so that the pupils will be aware of their application in other situations. When is the
Pigeon Hole Principle useful? When should counting techniques be used? Is now the time to
generalise? What happens if we work from the other end? Is there just one counter-example? Could the
computer do this for us? Type II activities should help students to learn the tricks of the mathematical
trade.
Type III
In Type III enrichment activities the student becomes an actual investigator of a real problem using
appropriate methods of inquiry. At this point students actually become mathematicians. They tackle a
mathematical problem using the techniques that a mathematician would employ. Preferably (and this
may be difficult to arrange) the problem they investigate is also a problem which they have taken an
active part in devising.
It is very unlikely that students will be able to take complete responsibility for generating a
problem which interests them. My experience is that (even at the Olympiad level) problem-producing
does not come easily. However, the teacher needs to be ready to take an idea and help the student
convert it into a viable problem.
This is not easy. After a while you develop an intuition for what might work, but you can
never be sure. Sometimes the problems are too easy. Sometimes they are impossibly hard, in which
case you have to learn to stop and go in another direction.
With original problems there is only one way to proceed: you have to jump in at the deep end.
Therefore wise teachers stand by with a life jacket, just in case. If the student isn’t able to swim it will
14
be necessary to start again on a subset of the problem in order to produce something which will lead to
a result.
Having produced the problem there may be no obvious technique to solve it. This is why Type
II activities exist – to give some new techniques which might be useful. However, intuition and
guesswork may be just as important.
Results
In a Type III activity it is important to communicate results. This should be more than putting the work
on the classroom noticeboard. Perhaps the proof or extension could appear in a school magazine or
newspaper. (After all, poems and stories appear there. Why not original maths?) A friendly academic
may be able to recommend somewhere else, and exceptional work might be published in, for example,
the New Zealand Mathematics Magazine or Function.
The attempt to publish or bring the result to the notice of an expert is important for three
reasons. Firstly, it increases the chances that the problem has been truly solved, as experts will review
it. Secondly, your solution may lead to new proofs and extensions by other people. Thirdly, and
perhaps just as important, it gives recognition to the work; it says, ‘Student, you have done well.’
So, following the three Types of activity Renzulli suggests, has led us to precisely what
mathematicians do. Mathematicians search for a problem. Then they seek for ways to solve it. Finally
they publish their result. Students can learn mathematics in this way too. In the process they will
experience the joy of creating their own mathematics. Hopefully this will be more interesting and lead
to better learning than the traditional algorithm learning approach to the subject.
5. Some Examples
Here are some examples of Type II activities developed into Type III activities. I hope you will see
that, by taking almost any situation that you have been familiar with for a long time, can be developed
into a Type III activity.
Exam ple 1
(1.1) Given a 3 litre jug and a 5 litre jug and a water trough, can you measure out exactly 4 litres
of water?
This is a fairly straightforward question that even young children can work at using trial and error. It
can be developed into a project by going to the next question.
(1.2) Given a 3 litre jug and a 5 litre jug, what amounts of water can you measure?
This question is amenable to a rigorous proof by 11- and 12-year-olds. But first the students have to do
some experimental work. (With pen and paper will do though it might be more fun with water.) The
experimental work can lead to a conjecture and then the conjecture will need to be proved or a counterexample found.
The answer here is that any whole number of litres can be produced. The proof is to first get 1 litre.
Once you have 1 litre you just keep adding 1 litre lots together and you eventually get any whole
number of litres. It’s important for students to
(a) get the idea of the proof, either through discussion in a group, or by their own individual
experimenting,
(b) write up the proof.
At this stage brainstorming should come up with a whole host of problems related to (1.1) and
(1.2). These will vary in sophistication depending on the maturity and ability of the students. With
older students you might get them inventing the following problem (Type III activity).
(1.3) Given an x litre, jug and a y litre jug, what amounts of water can you measure accurately?
A further extension: in (1.3) x and y will probably represent whole numbers.
(1.4) What if x and y are just any positive numbers?
15
Students should work on whichever problem they come up with and produce a proof which is
appropriate to their level of sophistication. For students who know the Euclidean Algorithm it is
relatively straightforward to show that if x and y have no common factors, then all whole numbers of
litres can be obtained.
(1.5) What if x and y have a greatest common factor of z though?
This activity can give students a chance to practice the creative side of mathematics. There is ample
opportunity for experimentation, conjecture and proof, while all the time using known algorithms. (If
an algorithm, such as the Euclidean Algorithm is needed but not known, then the students can be
directed to a book to learn it for themselves, or it can be taught. This is potentially a Type II activity.)
These activities can be used with small groups or a whole class. You could put the activities (1.1) to
(1.5) on cards and let students work on one card at a time. But this leaves no obvious chance for Type
III work.
How can it motivate a Type III activity? What possible problems come to mind as a result of
this example? Of course. it’s impossible to answer that question. It might be that the student goes for
rods, x and y units long, and asks what lengths can be made with rods of these lengths. Who knows
what they will come up with?
Exam ple 2
(2.1) In the diagram the circle centre A has radius 2cm and the circle centre B has radius lcm.
What is the radius of circle C?
This problem can be solved by an application or two of Pythagoras’ Theorem.
While not a strong Type II activity it does require some ingenuity to solve the problem. A
common error is to assume that there is a right angle at C. In fact no matter what the radii of the two
larger circles it can be shown that the ACB is never 90°. One nice problem is to explore the values of
ACB.
Brainstorm. What questions arise from (2.1)?
(2.2) Suppose the largest possible circle centre D is placed (squeezed) between the circles A and
C, and the horizontal line. What is the radius of D?
(2.3) Repeat (1.2). In other words, find the radius of the largest circle which lies between the
circles with centres A and D, and the horizontal line. In this way there are a whole series of
circles wedged in between the circle centre A, the horizontal line and the last circle. Find a
formula for the radii of all these.
Students should now try to develop their own Type III activity based on (2.1). Two possibilities
are suggested below but there are possibly infinitely many other questions that could be posed.
(2.4) Let P1 be the first circle (centre A) and P2 the second circle (centre B). Let P i be the largest
circle contained between the circles Pi – 1, Pi – 2 and the horizontal line, where i  3. Find a formula
for the radius of circle Pi.
(2.5) Let P1 = Q1, P2 = Q2 and P3 = Q3, where P1, P2, P3 are as in (2.4). Let Qi for i  4 be the
largest circle contained between Q1, Q2 and Qi – l. Find a formula for the radius of Qi.
Exam ple 3
(3.1) Call a number n good, if the numbers 1, 2, 3, …, n, can be split into two groups such that
the numbers in one group have the same sum as the numbers in the other.
(For students who know about sets, let N = {1, 2, 3, …, n}. We say that n is good if N can be divided
into two sets A and B such that AB = N, AB =  and the sum of the numbers in A is equal to the
sum of the numbers in B.)
16
This is a nice Type II exploration that has a simple solution. The proof that I know does
depend on knowing the sum of the numbers in N. However, a lot can be gained from this problem
without knowing this. Primary school students should be able to guess the pattern even if they can’t
prove that their conjecture is true.
Taking this activity on to Type III is possible, perhaps with several children brainstorming.
Perhaps they will break N into 3 sets to see what problems arise.
Exam ple 4.
(4.1) Now 9 = 4 + 5 and 9 = 2 + 3 + 4. So 9 can be written as the sum of a consecutive string of
positive integers in two ways. Are there any numbers that cannot be written as a consecutive
string of positive integers?
Experiment. Start with 1,2,3,4,5, … and so on. This is a classic chase. Do you see any patterns? Can
you make a conjecture? Can you prove the conjecture? All the good old stuff.
This problem like many others can be dealt with at a series of levels.
(i) Primary school students can, if nothing else, get good practice in arithmetic by looking for
examples of numbers which are not the sum of a consecutive string.
(ii) Many of these students can make a reasonable conjecture.
(iii) The proof is probably difficult to expect till secondary school. I have seen bright 10- and 11-yearolds get there with help.
(iv) What questions does this problem suggest? Brainstorming, may bring up questions that pupils (and
teachers) can’t solve. In that event, ring up a sympathetic academic.
Clearly the Type III suggestions I’ve made are not Type III in the strictest sense of the word because
they are not the student’s own problems. However, students will come up with questions that I haven’t
thought of. They do it all the time. They just need encouragement and help.
Notes
Dr Derek Holton is Professor of Mathematics and Statistics at the University of Otago, Box 56, Dunedin, New
Zealand.
He recommends a chapter by Alan Schoenfeld as reinforcing and extending the ideas in this set item. It is
Schoenfeld, Alan H. (1992) Learning to think Mathematically: Problem Solving, Meta Cognition, and SenseMaking in Mathematics, in, Grouws, Douglas (Ed) (1992) Handbook of Research Teaching and Learning,
New York: Macmillan.
References
Renzulli, J.S. (1977) The Enrichment Trial Model, Connecticut: Creative Learning Press.
Roe, Ann (1952) The Making of a Scientist, New York: Dodd and Mead.
Copying Perm itted
 Copyright on this item is held by NZCER and ACER who grant to all people actively engaged in education the
right to copy it in the interests of better teaching. Please acknowledge the source.
17
Poster 1
“CUTTING A SQUARE INTO 11 SQUARES”
2 sizes
3 sizes
(4 x 32, & 7 x 22)
(2 x 32, 3 x 22, & 6 x 12)
4 sizes
5 sizes
6 sizes
More than six sizes
(1 x 72, 2 x 52, 1 x 42, 2 x 32, 2 x 22 & 3 x 12)
18
The Open University
ISBN 0 7492 5792 X